Coordinate transformation apparatus



June 25, 1963 A. M. LOSHIN 3,094,783

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June 25, 1963 A. M. LOSHIN 3,094,783

COORDINATE TRANSFORMATION APPARATUS Filed Oct. 2, 1957 3 Sheets-Sheet 3 INVENTOR.

United States Patent 1 3,094,783 COORDINATE TRANSFORMATION APPARATUS Albert M. Loshin, 8640 Bay Parkway, Brooklyn 14, N .Y. Filed Oct. 2, 1957, Ser. No. 687,844 1 Claim. (Cl. 33-24) (Granted under Title 35, U.S. Code (1952), sec. 266) The invention described herein may be manufactured and used by or for the United States Government for governmental purposes without payment to me of any royalty thereon.

The invention relates generally to a method and means for reproducing figures such as plots of orthogonal graphs and at the same time varying the ratio of the horizontal and vertical dimensions. More particularly, the invention concerns a simple and direct technique whereby figures including plots of orthogonal graphs can be reproduced in such a manner that the relationship of hori zontal to vertical dimensions can be changed to a new and desired ratio.

In the reproduction of graphs for analysis and comparison, a problem arises when the scales and dimensions vary on the graphs or figures to be compared. Also in the publication of prepared graphs a similar problem is encountered when it is desired to use .a certain plot in a particular limited space of a definite size and shape in the printed report. The present invention overcomes these dilficulties by presenting relatively low cost easyto-use methods for varying the relative dimensions of plots, graphs, drawings or other figures by a process of reproduction thereby eliminating the necessity of tedious and time-consuming procedure of redrawing or replotting the figures in order to obtain the desired size.

Accordingly, it is a primary object of the invention to provide .a method and means for reproducing graphs and drawings whereby either the vertical or horizontal dimension or both can be varied independently so that the resulting reproduction is of some desired shape and size or some new and different ratio of height to width.

Another object of the invention is to provide relatively simple and easy-to-use apparatus for accomplishing the desired ratio variation of the figure to be reproduced. The apparatus mechanically, photographically, or otherwise projects a given figure onto an angularly related plane thus effecting the desired change in dimensions propor tional to the angular difference between the plane on which the original figure is mounted and the inclined copy board.

A particular advantage of this method of reproduction lies in the fact that by determining the angle between the planes, the new ratio can easily be computed because there exists the simple relationship of 1: cos 6, where 0 is the angle between the planes in singly oblique projection. In double oblique projection where it is desired to vary both dimensions, the angle 6 is measured from a reference plane which lies perpendicular to the direction of projection.

These and other objects, features, and advantages will become more apparent after considering the following detailed description taken in connection with the annexed drawings and appended claim.

In the drawings:

FIG. 1 illustrates a typical curve to be varied;

FIG. 2. shows a view of the curve having the desired height to width ratio;

FIG. 3 shows a schematic view of the transformation;

FIG. 4 shows schematically an example of doubly oblique projection;

FIGS. 5 and 6 depict a mechanical device for producing the transformation; and

FIGS. 7 and 8 show photographic apparatus for accomplishing this result.

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Referring now to FIG. 1, there is shown a curve 12 plotted and drawn on graph paper. For the sake of simplicity assume that the figure is of essentially equal height and width, that is, the x and y axes are equal. Then, suppose it is desirable to transform this curve 12 to a curve 14 shown in FIG. 2, where the y or horizontal dimension is 1.41 times longer than that of the x or vertical dimension.

To determine the angular inclination of copy plane 18 in FIG. 3 with relation to the reference plane 16 in order to accomplish the desired ratio change, it is necessary to find the angle 0. Since the relationship of the two curves vary as the cosine of this angle then the required angle would be the one whose cosine was equal to or approximately 45. Then, by inclining the copy plane 18, 45 in relation to the reference plane 16 and projecting the curve 12 onto the copy plane, the resulting curve 14 will be increased in the horizontal or y direction by a factor of of 1.41, while the vertical or x dimension remains the same.

It may be desirable in certain instances to simultaneously decrease one of the dimensions while increasing the other. In FIG. 4 there is shown an example of this particular variation. Using the same curve 12 as the original, the copy plane 18 is inclined at 45 thus the horizontal dimension is increased the same amount as in FIG. 3. However, since the original is not mounted on the reference plane 16 but, instead, is on a different plane 20 placed at an angle 0', 30 from the reference plane 16. Since the cosine of 30 is .866, then the vertical dimension of the curve 12 would be reduced by this factor resulting in a vertical dimension ratio of 1:.866 while the horizontal dimension ratio has increased to 1:1.41. This process may be called doubly oblique projection and results in a curve 22 which has changed from a horizontal to vertical ratio of 1:1 to a ratio of 1.41:.866.

This invention is particularly concerned with the method and means for accomplishing the projection of the original curve in such a manner as to result in the desired variation. A mechanical device having a pantograph linkage system for projecting a curve is shown in FIG. 5 with an exploded detail of some of its parts in FIG. 6.

The device is supported on a stable base 31 on which the frame and most of the supports are directly mounted. The principal frame member 32 serves as a support for the original-figure holder 33 which is located some distance above the base 31. A side frame 35 is employed to support the varied-figure holder 37 and since these two holders must be capable of being rotated or tilted with respect to one another, provision is made to make the varied-figure holder 37 singly oblique in respect to the original-figure holder 33. This is effected by mounting the varied-figure holder 37 on the side frame 35 with hinges 36.

Provision is made to lock said holder 37 in any desired position by a slotted and calibrated support member 39 and a position locking knob 40. Additional rigidity of the varied-figure holder 37 can be obtained, if necessary, by adjusting and locking one or more adjustable spot supports 41 which might be included for this purpose.

A flange 42 fixed to the base member 31 supports the principal supporting rod 43 which in turn supports the entire tracing apparatus. Two clamps 44 locked in place by suitable bolts, serve to mount the tracing apparatus on the principal supporting rod 43. These clamps 44 each hold a vertical cylindrical bushing 46 shown in detail in FIG. 6. The swiveling rod 45 is mounted in these bushings and, as its name implies, turns freely around on its axis thus acting as a pivot forthe remainder of the tracing apparatus. The pivot-guide rod connector 47, shown in detail in the FIG. 6, connects the substantially horizontal guide rods 48 to the swiveling rod 45 and also restrains them from moving up and down when the clamps 44 are properly spaced. The guide rod connectors 47 should ride on the ends of the cylindrical bushings 46 with sliding contact, lubricated to reduce friction and wear.

The guide rods 48 rotate in one plane only and restrain the styli riders 50 and 54 to a single degree of freedom, namely along the axis of said guide rods 48. The substantially horizontal guide rods 48 are given rigidity by joining their ends with two substantially vertical connecting rods 49, thus forming a complete structure loop. The tracing stylus rider 50 is free to move in a plane to permit tracing a curve. The writing stylus rider 54 identically follows the movement of the tracing stylus rider S in a different plane. This will permit the tracing of any point on the figure, and locates such point by what is tantamount to its polar coordinates with respect to the pivot.

T he two riders 50 and 54 are to have the same identical motion. In order for the writing stylus rider 54 to be laterally displaced along the guide rods as the tracing stylus rider 50 is moved, these two riders must be rigidly connected. This is accomplished by the use of control rods 51, control-follower plate 52, and the follower rods 53. The control-follower plate 52 also rides on the guide rods 48 to give the structure greater rigidity.

To effect the actual projection, vertical movement must be allowed so that the vertical writing stem 56 will contact, at all times, the oblique plane of the varied-figure holder 37. This is accomplished by letting the rider 54 also act as a bushing for the vertical writing stem 56. This stem 56, which is free to move up and down, is terminated with a writing stylus holder 57, whose writing point is kept against the varied figure by gravity. A weight 55 is added to the upper end of the writing stem 56 to insure that the Writing stylus will follow the oblique converted-figure plane, and to supply sufficient writing pressure. A partially completed converted figure 38, and its associated original figure 34 are shown mounted on their respective holders 37 and 33. In actual use, the stylus rider 50 would be moved manually keeping the tracing stylus along the original figure 34 and simultaneously the writing stylus 54 by means of its Writing point will form the converted figure comparable to the original figure except that one of its orthogonal components is elongated.

The results as outlined above may also be obtained by the use of photographic means. FIG. 7 depicts apparatus for accomplishing the projection where the light source is immediate or close to the original figure.

Using the photographic method it is necessary to obtain a parallel beam of light. This may be accomplished as shown in FIG. 7 by providing a point source of light 60 located at the focus of parabolic reflector 61. A nonrefiective shield 62 may be used to prevent a photographic hot-spot or high intensity spot of light from existing, and to eliminate direct radiation which would be radial .rather than parallel. The parabolic reflector 61 has a flanged non-reflecting rim 63 which is padded with felt 64 or some other material so as not to harm the film 65. Ordinarily said film 65 would be a negative of the original figure to be transformed.

A flat transparent plate of glass66 can be used to sup- .port the original film 65. Another flat plate of glass (not shown) may be placed on top to protect and/or flatten the film 65. The source of parallel light may be scanned ,or moved in the plane of the original film if it is not sufficiently large or uniform in intensity to cover the entire film. The original figure on film 65 is projected on to an oblique plane 68 located on the other side of the original film, producing a latent image on the unexposed photosensitive paper or film 69 which lies on the oblique plane 68. A frame 70 with flat cover glass 67 might be used 4 to advantage in framing the unexposed paper 69 so that unwanted distortion will be avoided. The light source 60 is kept lighted for a time adequate to permit proper exposure.

In FIG. 8, which is an example of doubly oblique pro- 'ection, a remote source of parallel light is specified. Since two transformations are possible in this case, the parallel source of light is not necessarily perpendicular to the plane on which the original film is held. It is, however, parallel with the plane of 'base and perpendicular to the common edge between 75 and plane 76, said planes being attached at their common edges by hinges 77. Since the parallel beam is quite remote only a portion of it is utilized, hence, would be less critical as to evenness of intensity.

The device as shown in FIG. 8 consists of a base 75 onto which a front frame 76 is attached by means of hinges 77. An angular scale 78 is provided to hold said frame 76 at the desired angle and to indicate said angle in relation to the base 75 or calibrated directly in accordance with the corresponding transformation ratio. Clamps 79 are provided to hold the original film or plate 80 in its proper position. The rear frame 81 is attached by means of binge 82 to upright member 83 which in turn is firmly attached to the base 75. The angular scale 84 is provided so that the angular displacement of the rear frame 81 can be determined and also to hold said frame at this desired angle. Clamps (not shown) are provided to hold the photosensitive paper or film 85 in place for exposure. This device can simultaneously reduce the axis perpendicular to the base and increase the axis parallel with the base. Such a double transformation is seldom warranted except for extending the possible range of ratio changes, particularly Where the film grain would be the limiting factor.

Although the invention has been described with reference to particular embodiments, it will be understood to those skilled in the art that various changes and modi fications can be made therein without departing from the invention.

Having thus described my invention, what I claim as new and desire to secure by Letters Patent of the United States is:

A mechanical transformation apparatus for reproducing plane figures by tracing while concurrently varying the ratio of the height-to-width dimensions thereof, said apparatus comprising:

(a) an elongated member,

(b) means mounting said member for rotation about a first axis transverse to the longitudinal axis of said member so that said member is arranged to experience angular motion about said first axis,

(0) a pair of riders located at different positions relative to the length of the first axis and mounted on said member to slide lengthwise along the member,

said riders being coupled to one another for joint lengthwise movement along said member to maintain them at equal radial distances from the first axis so that both riders are arranged to experience in two parallel planes identical polar motions about said first axis, V

(d) a tracing stylus carried by one rider,

(:2) a writing stylus carried by the other rider,

(1) means mounting at least one stylus on its rider for free movement relative thereto along an axis parallel to the first axis, 1

(g) a first plane support for an original figure to be traced,

(h) a second plane support for the traced figure,

(i) one of said plane supports being located below and in operational association with the tracing stylus and the other of said plane supports being located below and in operational association with the-writing stylus,

(j) and means mounting said supports in planes transverse to said first axis and including means to selectively adjustably change relative to said first axis the angular position of at least the plane of the support operatively associated with the stylus that is freely movable parallel to the first axis, (k) said last-named stylus being biased toward its associated plane support, (I) whereby the height-to-width of the reproduced figme can be selectively diflered from the height-towidth ratio of the original figure by varying the angu- 10 13.1" position of the plane of the adjustable support.

References Cited in the file of this patent UNITED STATES PATENTS Bartholowsky Feb. 18, 1936 Wellington Mar. 2, 1937 Davidson et a1. Sept. 24, 1940 McQuaid Jan. 11, 1944 Wood Oct. 15, 1946 Ivy Apr. 15, 1947 Merriam June 30, 1953 Deak'in May 22, 1956 

